Properties

Label 2-1690-13.9-c1-0-12
Degree $2$
Conductor $1690$
Sign $-0.979 - 0.202i$
Analytic cond. $13.4947$
Root an. cond. $3.67351$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.366 + 0.633i)3-s + (−0.499 − 0.866i)4-s − 5-s + (−0.366 − 0.633i)6-s + (1.5 + 2.59i)7-s + 0.999·8-s + (1.23 + 2.13i)9-s + (0.5 − 0.866i)10-s + (−1.5 + 2.59i)11-s + 0.732·12-s − 3·14-s + (0.366 − 0.633i)15-s + (−0.5 + 0.866i)16-s + (4.09 + 7.09i)17-s − 2.46·18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.211 + 0.366i)3-s + (−0.249 − 0.433i)4-s − 0.447·5-s + (−0.149 − 0.258i)6-s + (0.566 + 0.981i)7-s + 0.353·8-s + (0.410 + 0.711i)9-s + (0.158 − 0.273i)10-s + (−0.452 + 0.783i)11-s + 0.211·12-s − 0.801·14-s + (0.0945 − 0.163i)15-s + (−0.125 + 0.216i)16-s + (0.993 + 1.72i)17-s − 0.580·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 - 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1690\)    =    \(2 \cdot 5 \cdot 13^{2}\)
Sign: $-0.979 - 0.202i$
Analytic conductor: \(13.4947\)
Root analytic conductor: \(3.67351\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1690} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1690,\ (\ :1/2),\ -0.979 - 0.202i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.150144202\)
\(L(\frac12)\) \(\approx\) \(1.150144202\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + T \)
13 \( 1 \)
good3 \( 1 + (0.366 - 0.633i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-4.09 - 7.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.232 - 0.401i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.73 + 8.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.26 + 2.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.73T + 31T^{2} \)
37 \( 1 + (-0.401 + 0.696i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.19 - 9i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1 - 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 - 0.464T + 53T^{2} \)
59 \( 1 + (-5.19 - 9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.09 + 5.36i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 + 4.19T + 79T^{2} \)
83 \( 1 + 8.19T + 83T^{2} \)
89 \( 1 + (3.40 - 5.89i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.56 + 7.90i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.836721962958712620302869559964, −8.549843226020884942595296672672, −8.275982182564330186224569780366, −7.50346360922747266438279866846, −6.51194614666180290725838577816, −5.62867957331346145115317194941, −4.84253105455964250925700708913, −4.25096091262461337598262828267, −2.70069514871009343499847631377, −1.51844330631269203433712910445, 0.58730806193021268247447899331, 1.32657033784862986921348477638, 3.01904704863826243820153448571, 3.64756303207757472384509125414, 4.76062182357313423418662933609, 5.57545999904518620816433855805, 7.03993958539146011846096454537, 7.32249585617473071072743575560, 8.125494581801603011735228793132, 9.079594708557800673698469945785

Graph of the $Z$-function along the critical line